Negative Quadrant Dependent Random Variables Research Articles

NEW CONCENTRATION INEQUALITIES AND COMPLETE CONVERGENCE FOR ELNQD RANDOM VARIABLES WITH APPLICATION TO LINEAR MODELS GENERATED BY ELNQD ERRORS

In this paper, we introduce the concept of extended linear negative quadrant dependence (ELNQD, in short). We establish a new concentration inequalities and complete convergence for the distribution of sums of extended linear negative quadrant dependent random variables. Using these inequalities for proved the complete convergence of first autoregressive processes model generated by identically distributed ELNQD errors.

Probability tail for linearly negative quadrant dependent random variables of partial sums and application to linear model

In this paper, we establish a new concentration inequality and complete convergence of weighted sums for arrays of rowwise linearly negative quadrant dependent (LNQD, in short) random variables and obtain a result dealing with complete convergence of first-order autoregressive processes with identically distributed LNQD innovations.

On the linearly extended negative quadrant dependent random variables and its inequalities

In this article, the structure of linearly extended negative quadrant dependent (LENQD) random variables is given, and its basic properties are investigated. In addition, the characteristic function inequality, Bernstein-type inequality, some probability inequalities, and the Rosenthal-type inequality are established for LENQD random variables.

On the convergence for weighted sums of Hilbert-valued coordinatewise pairwise NQD random variables and its application

In this article, we investigate complete convergence and strong laws of large numbers for weighted sums of coordinatewise pairwise negative quadrant dependent random variables taking values in Hilbert spaces. As an application, the complete convergence and the almost sure convergence of degenerate von Mises-statistics are investigated.

NEW EXPONENTIAL PROBABILITY INEQUALITY AND COMPLETE CONVERGENCE FOR F-LNQD RANDOM VARIABLES SEQUENCE WITH APPLICATION TO AR(1)MODEL GENERATED BY F-LNQD ERRORS

The exponential probability inequalities have been important tools in probability and statistics. In this paper, we prove a new tail probability inequality for the distri-butions of sums of conditionally linearly negative quadrant dependent (F-LNQD , in short) random variables, and obtain a result dealing with conditionally complete con-vergence of first-order autoregressive processes with identically distributed (F-LNQD) innovations.

On the convergence for PNQD sequences with general moment conditions

Let {X, Xn, n ≥ 1} be a sequence of identically distributed pairwise negative quadrant dependent (PNQD) random variables and {an, n ≥ 1} be a sequence of positive constants with an = f(n) and f(θk)/f(θk−1) ≥ β for all large positive integers k, where 1 0 (x ≥ 1) is a non-decreasing function on [b, +∞) for some b ≥ 1. In this paper, we obtain the strong law of large numbers and complete convergence for the sequence {X, Xn, n ≥ 1}, which are equivalent to the general moment condition $$\sum\nolimits_{n = 1}^\infty {P\left( {\left| X \right| > {a_n}} \right) < \infty }$$. Our results extend and improve the related known works in Baum and Katz [1], Chen at al. [3], and Sung [14].

A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

By using some inequalities for linearly negative quadrant dependent random variables, Berry-Ess$\acute{e}$en bound of wavelet estimation for a nonparametric regression model is investigated under linear process errors based on linearly negative quadrant dependent sequence. The rate of uniform asymptotic normality is presented and the rate of convergence is near $O(n^{-\frac{1}{6}})$ under mild conditions, which generalizes or extends the corresponding results of Li et al.(2008) under associated random samples in some sense.

Convergence in $$\varvec{p}$$ p -Mean for Arrays of Random Variables

In this paper, conditions are given to ensure the convergence in mean of order $$p (1 \leqslant p < 2)$$ for arrays of random variables. Recent results about convergence in p-mean for arrays of row-wise pairwise negative quadrant dependent random variables will be improved.

Some mean convergence theorems for arrays of rowwise pairwise negative quadrant dependent random variables

For arrays of rowwise pairwise negative quadrant dependent random variables, conditions are provided under which weighted averages converge in mean to 0 thereby extending a result of Chandra, and conditions are also provided under which normed and centered row sums converge in mean to 0. These results are new even if the random variables in each row of the array are independent. Examples are provided showing (i) that the results can fail if the rowwise pairwise negative quadrant dependent hypotheses are dispensed with, and (ii) that almost sure convergence does not necessarily hold.

Uniformly asymptotic normality of sample quantiles estimator for linearly negative quadrant dependent samples

In the present article, by utilizing some inequalities for linearly negative quadrant dependent random variables, we discuss the uniformly asymptotic normality of sample quantiles for linearly negative quadrant dependent samples under mild conditions. The rate of uniform asymptotic normality is presented and the rate of convergence is near O(n^{-1/4} log n) when the third moment is finite, which extends and improves the corresponding results of Yang et al. (J. Inequal. Appl. 2011:83, 2011) and Liu et al. (J. Inequal. Appl. 2014:79, 2014) under negatively associated random samples in some sense.

Some limit theorems for weighted negative quadrant dependent random variables with infinite mean

In the present paper, we will investigate weak laws of large numbers for weighted pairwise NQD random variables with infinite mean. The almost sure upper and lower bounds for a particular normalized weighted sum of pairwise NQD nonnegative random variables are established also.

English

The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables”.

Maximal inequalities and some convergence theorems for fuzzy random variable

Some maximal inequalities for quadratic forms of independent and linearly negative quadrant dependent fuzzy random variables are established. Strong convergence of such quadratic forms are proved based on the martingale theory. A weak law of large numbers for linearly negative quadrant dependent fuzzy random variables is stated and proved.

Further study of complete convergence for weighted sums of PNQD random variables

Based on Wu (J. Appl. Math. 2012:104390, 2012), the complete convergence for weighed sums of pairwise negative quadrant dependent (PNQD) random variables is further studied under weaker weighted condition. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965), Wu (J. Appl. Math. 2012:104390, 2012) and Zhang (J. Inequal. Appl. 2014:353, 2014).

Complete convergence theorems for normed row sums from an array of rowwise pairwise negative quadrant dependent random variables with application to the dependent bootstrap

Let {X n,j , 1 ⩽ j ⩽ m(n), n ⩾ 1} be an array of rowwise pairwise negative quadrant dependent mean 0 random variables and let 0 < b n → ∞. Conditions are given for $$\sum\nolimits_{j = 1}^{m(n)} {{X_{n,j}}/{b_n} \to 0} $$ completely and for $${\max _{1 \leqslant k \leqslant m(n)}}|\sum\nolimits_{j = 1}^k {{X_{n,j}}|/{b_n} \to 0} $$ completely. As an application of these results, we obtain a complete convergence theorem for the row sums $$\sum\nolimits_{j = 1}^{m(n)} {X_{n,j}^*} $$ of the dependent bootstrap samples $$\{ \{ X_{n,j}^*,1 \leqslant j \leqslant m(n)\} ,n \geqslant 1\} $$ arising from a sequence of i.i.d. random variables {X n , n ⩾ 1}.

Complete convergence for weighted sums of LNQD random variables

In this paper, we study the complete convergence for weighted sums of linearly negative quadrant dependent (LNQD) random variables based on the exponential bounds. In addition, we present some complete convergence for arrays of rowwise LNQD random variables.

Complete moment convergence of pairwise NQD random variables

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.

On the Exponential Inequality for Weighted Sums of a Class of Linearly Negative Quadrant Dependent Random Variables

The exponential inequality for weighted sums of a class of linearly negative quadrant dependent random variables is established, which extends and improves the corresponding ones obtained by Ko et al. (2007) and Jabbari et al. (2009). In addition, we also give the relevant precise asymptotics.

Strong Limit Theorems for Pairwise NQD Random Variables

A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem. Our result generalizes and improves on the corresponding one in Li and Yang (2008). We also obtain a complete convergence result for an array of rowwise pairwise NQD random variables.

Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables

Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables